I've bought both ordinary glasses and prescription sunglasses in Singapore several times. You do have to be a little bit careful in that you get what your prescription says and I'd actually recommend getting your eyes tested at home and taking that prescription with you (if you can do this in the US - you can in the UK).

But as for choosing them on one visit and collecting them on the next, should be no problem.

Can't remember where we bought them from when we've been there but we've bought from more than one place. And they are cheaper than the UK. Not sure what glasses cost in the US so not sure about the cheapness related to there.